Remarks on generalized weak KKM multimaps

Year 2022, Volume: 5 Issue: 1, 1 - 11, 31.03.2022

Sehie Park

https://doi.org/10.53006/rna.1055249

Cited By: 3

Abstract

The concept of generalized KKM maps was initiated by Kassay-Kolumbán in 1990 and Chang-Zhang in 1991. Recently, Balaj and his colleagues extended generalized KKM maps w.r.t. a multimap to weak KKM maps and generalized weak KKM maps w.r.t. a multimap and applied them to various problems in the KKM theory. However, their results are mainly concerned within the realm of topological vector spaces. Our aim in this article is to extend some of them to abstract convex spaces. Some related facts are also discussed.

Keywords

Abstract convex space, KKM maps, weak KKM map, generalized weak KKM map, generalized weak KKM map, generalized KKM map

References

• [1] Agarwal, R.P., Balaj, M., O’Regan, D. Variational relation problems in a general setting, J. Fixed Point Theory Appl. 18 (2016) 479–493.
• [2] Agarwal, R.P., Balaj, M., O’Regan, D. Common fixed point theorems in topological vector spaces via intersection theorems, J. Optim. Theory Appl. 173 (2017) 443–458.
• [3] Agarwal, R.P., Balaj, M., and O’Regan, D. Intersection theorems with applications in optimization, J. Optim. Theory Appl. 179 (2018) 761–777.
• [4] Agarwal, R. P., Balaj, M., and O’Regan, D. Intersection theorems for weak KKM set-valued mappings in the finite-dimensional setting, Top. Appl. 262 (2019) 64–79.
• [5] Balaj, M. Weakly G-KKM mappings, G-KKM property, and minimax inequalities, J. Math. Anal. Appl. 294(1) (2004) 237–245.
• [6] Balaj, M. A common fixed point theorem with applications to vector equilibrium problems, Appl. Math. Lett. 23 (2010) 241–245.
• [7] Balaj M. Intersection theorems for generalized weak KKM set-valued mappings with applications in optimization, Math. Nach. (2021) 1–15.
• [8] Balaj M. Existence results for quasi-equilibrium problems under a weaker equilibrium condition, Operations Research Letters 49 (2021) 333–337.
• [9] Ben-El-Mechaiekh, H., Deguire, H.P. and Granas, A. Points fixes et coincidences pour les applications multivoques, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982) 337–340.
• [10] Ben-El-Mechaiekh, H., Deguire, H.P. and Granas, A. Points fixes et coincidences pour les fonctions multivoques, II, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982) 381–384.
• [11] Chang, S.-S. and Zhang, Y. Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl. 159 (1991) 208–223.
• [12] Granas, A. and Liu, F.C. Coincidences for set-valued maps and minimax inequalities, J. Math. Pures Appl. (9) 65 (1986) 119–148.
• [13] Ha, C.W. Minimax and fixed point theorems, Math. Ann. 248 (1980) 73–77.
• [14] Ha, C.W. A minimax theorem, Acta Math. Hung. 101 (2003) 149–154.
• [15] Jeng, J.C., Hsu, H.C., and Huang, Y.Y. Fixed point theorems for multifunctions having KKM property on almost convex sets, J. Math. Anal. Appl. 319 (2006) 187–198.
• [16] Kassay, G. and Kolumbán, I. On the Knaster-Kuratowski-Mazurkiewicz and Ky Fan’s theorem, Babes-Bolyai Univ. Res. Seminars Preprint 7 (1990) 87–100.
• [17] Kim, H. and Park, S. Generalized KKM maps, maximal elements and almost fixed points, J. Korean Math. Soc. 44 (2007) 393–406.
• [18] Lee, W. A remark on generalized KKM maps on KKM spaces, Nonlinear Anal. Forum 21(1) (2016) 1–5.
• [19] Lin, L.J., Ko, C.J. and Park, S. Coincidence theorems for set-valued maps with G-KKM property on generalized convex space, Discuss. Math. Differential Incl. 18 (1998) 69–85.
• [20] Luc, D. T. An abstract problem in variational analysis, J. Optim. Theory Appl. 138 (2008), 65–76.
• [21] Park, S. Remarks on weakly KKM maps in abstract convex spaces, Inter. J. Math. Math. Sci., vol.2008 (2008), Article ID 423596, 10 pp. doi:10.1155/2008/423596.
• [22] Park, S. The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010)1028–1042.
• [23] Park, S. Evolution of the 1984 KKM theorem of Ky Fan, Fixed Point Theory Appl. vol.2012, 2012:146. DOI:10.1186/1687-1812-2012-146.
• [24] Park, S. A unified approach to generalized KKM maps, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 55(1) (2016) 1–20.
• [25] Park, S. Various examples of the KKM spaces, J. Nonlinear Convex Anal. 21(1) (2020) 1–19.
• [26] Park, S. Extending KKM theory to a large scaled logical system, [H.-C. Lai Memorial Issue] J. Nonlinear Convex Anal. 22(6) (2021) 1045–1055.
• [27] Park, S. and Lee, W. A unified approach to generalized KKM maps in generalized convex spaces, J. Nonlinear Convex Anal. 2 (2001) 157–166.